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Full text of "Treatise On Analysis Vol-Ii"

9   MEASURABLE FUNCTIONS       139
A function/: X -> K is a step function if it takes only & finite number of

n

values ak (1 ^ k g n). In that case we have/= ] ak cpAk, where Ak =f~1(ak)9

with the convention that if ak is  oo, the value of ak(pAk is aA in Ak and 0 in
X  Ak (13.11). It follows immediately from this and from (13.9.9) that/is
measurable if and only if each of the Ak is measurable.

(13.9.12) Le/:X-R be measurable. Then there exists a sequence (gn)
of universally measurable step functions with compact support, such that
\gn(x)\ < \f(x)\ for all x e X and such that the sequence (gn(x)} converges al-
most everywhere to f(x) (which implies that f is equivalent to a universally
measurable function).

Consider a partition of X consisting of a sequence (Kn) of compact sets
and a negligible set N. For each integer i  n, there exists a finite covering
(Uij\jqtn of K by sets which are open in Ki9 such that the oscillation of
/in each U$ is g l/n (with respect to a distance on K, cf. (3.16.5)). Now we
have the following lemma:

(13.9.12.1) Let (Fh\^h^p be a finite family of integrable sets. Then there
exists a finite family (Gk)igkgr of pairwise disjoint integrable sets, such that
each Fh is the union of some of the Gk.

p
Consider the 2P - 1 sets of the form (| Zh, where each Zh is either F,, or

X  Fh, and Zh = F,, for at least one index h. Then the family of distinct non-
empty sets of this form satisfies the conditions of the lemma, because these sets

p
are integrable by (13.7.6) and each F/, is the union of the sets p) Zz for which

By applying this lemma to each of the finite families (^Jij\^j^qin9 we
obtain a. partition (A^X^^,.^ of Kf into universally measurable, integrable
sets. Let

0               if /changes sign in A^,

if/XAJJ?)        if /^O   on   A^,
[sup/(Ag>)      if /^O   on   Ag>,

and define gn as follows: gn(x) = a$ if x e A$ (1 ^ i 5* n, 1 < k ^ rin), and
#(*) == 0 otherwise. Then it is clear that the functions gn have the required
properties.ions obtained from semi-continuous